Halszka Tutaj-Gasińska

Counterexamples to the I(3)⊂I2 containment

We show that in general the third symbolic power of a radical ideal of points in the complex projective plane is not contained in the second usual power of that ideal. This answers in negative a question asked by Huneke and generalized by Harbourne.

Counterexamples to the

We show that in general the third symbolic power of a radical ideal of points in the complex projective plane is not contained in the second usual power of that ideal. This answers in negative a question asked by Huneke and generalized by Harbourne.

I^{(3)} \subset I^2

Abstract Inspired by results of Guardo, Van Tuyl and the second author for lines in P 3 , we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r-dimensional planes in P n for n ⩾ 2 r + 1 . These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P 2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.

containment

We study linear series on a projective plane blown up in a bunch of general points. Such series arise from plane curves of fixed degree with assigned fat base points. We give conditions (expressed as inequalities involving the number of points and the degree of the plane curves) on these series to be base point free, i.e. to define a morphism to a projective space. We also provide conditions for the morphism to be a higher order embedding. In the discussion of the optimality of obtained results we relate them to the Nagata Conjecture expressed in the language of Seshadri constants and we give a lower bound on these invariants.

Linear subspaces, symbolic powers and Nagata type conjectures

Abstract Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3] , [7] , [13] , [16] , [18] , [19] , [20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3] , [15] . There have been exciting new developments in this area recently. It had been expected for several years that I ( N r − N + 1 ) ⊆ I r should hold for the ideal I of any finite set of points in P N for all r > 0 , but in the last year various counterexamples have now been constructed (see [11] , [17] , [8] ), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

General blow-ups of the projective plane

Abstract In 1939 Keller conjectured that any polynomial mapping ƒ : C n → C n with constant nonvanishing Jacobian determinant, should be invertible. This open problem bears the name of Jacobian conjecture. Druzkowski proved that cubic linear mappings are sufficient to decide the conjecture. For this important class we develop an algorithm that translates the constant-Jacobian condition into algebraic equations in the matrix of parameters. We also single out a natural special case ot these conditions, that we call D-nilpotency. The class of D-nilpotent matrices turns out to coincide with set of matrices that are permutation-similar to upper-triangular matrices. The corresponding cubic-linear maps are always invertible.

Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

In the note we investigate in how many points can ℙ2 be blown up to have the pullback bundle k-very ample in a given 0-dimensional subscheme Z.

Drużkowski matrix search and D-nilpotent automorphisms

Starting with the pioneering work of Ein and Lazarsfeld [9] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [2,5,10,12,18,20,22,24]. In the present note we show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces [13,14] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number

Embeddings of Blown-Up Plane

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Restrictions on Seshadri constants on surfaces

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